For the past few days I’ve been pondering the question of how best to respond to paulfchristiano’s recentpostsandcomments questioning the value of mathematical research. I don’t think I can do it concisely, in a single post; bridging the inferential distance may require something more like a sequence of posts. I may end up writing such a sequence eventually, since it would involve ideas I’ve actually been wanting to write up for some time, and which are actually relevant to more than just the specific questions at issue here (whether society should sponsor mathematics, and given that it does, whether paulfchristiano or anyone else in the LW readership should pursue it).
However, as the preceding parenthetical hints at, I’m actually somewhat conflicted about whether I should even bother. Although I believe that mathematical research should be conducted by somebody, it’s not at all clear to me that the discipline needs more people beyond those who already “get” its importance, and are out there doing it rather than writing skeptical posts like paulfchristiano’s. It seems perfectly plausible to me that those who feel as paulfchristiano does should just leave the profession and do something else that feels more “important” to them. This is surely the best practical solution on an individual level for those who think they have a better idea than existing institutions of where the most promising research directions lie, at least until Hansonian prediction markets are (ever) implemented.
Nevertheless, for those interested in the society-level question of whether mathematics (as such) may be justifiably pursued by anyone, or any community of people, as a professional occupation (which is quite distinct from the question of whether e.g. paulfchristiano should personally pursue it), I recommend, at least as a start, grappling with the arguments put forward by the best mathematicians in their own words. I think this essay by Timothy Gowers (a Fields Medalist), titled “The Importance of Mathematics”, is a good place to begin. I would particularly draw the attention of those like paulfchristiano, who think they have a good idea of which branches of mathematics are useful and which aren’t, to the following passage, from pp.8-9 (unfortunately the illustrations are missing, but the point being made is pretty clear nonetheless):
So—mathematicians can tell their governments—if you cut funding to pure mathematical research, you run the risk of losing out on unexpected benefits, which historically have been by far the most important.
However, the miserly finance minister need not be convinced quite yet. It may be very hard to identify positively the areas of mathematics likely to lead to practical benefits, but that does not rule out the possibility of identifying negatively the areas that will quite clearly be useless, or at least useless for the next two hundred years. In fact, the finance minister does not even need to be certain that they will be useless. If a large area of mathematics has only a one in ten thousand chance of producing economic benefit in the next fifty years, then perhaps that at least could be cut.
You will not be surprised to hear me say that this policy would still be completely misguided. A major reason, one that has been commented on many times and is implied by the subtitle of this conference, “A Celebration of the Universality of Mathematical Thought”, is that mathematics is very interconnected, far more so than it appears on the surface. The picture in the back of the finance minister’s mind might be something like Figure 4. According to this picture, mathematics is divided into several subdisciplines, of varying degrees of practicality, and it is a simple matter to cut funding to the less practical ones.
A more realistic picture, though still outrageously simplified, is given in Figure 5. (Just for the purposes of comparison, Figure 6 shows Figures 4 and 5 superimposed.) The nodes of Figure 5 represent small areas of mathematical activity and the lines joining them represent interrelationships between those areas. The small areas of activity form clusters where there are more of these interrelationships, and these clusters can perhaps be thought of as subdisciplines. However, the boundaries of these clusters are not precise, and many of the interrelationships are between clusters rather than within them.
In particular, if mathematicians work on difficult practical problems, they do not do so in isolation from the rest of mathematics. Rather, they bring to the problems several tools - mathematical tricks, rules of thumb, theorems known to be useful (in the mathematical sense), and so on. They do not know in advance which of these tools they will use, but they hope that after they have thought hard about a problem they will realize what is needed to solve it. If they are lucky, they can simply apply their existing expertise straightforwardly. More often, they will have to adapt it to some extent
(...)
Thus, a good way to think about mathematics as a whole is that it is a huge body of knowledge, a bit like an encyclopaedia but with an enormous number of cross-references. This knowledge is stored in books, papers, computers and the brains of thousands of mathematicians round the world. It is not as convenient to look up a piece of mathematics as it is to look up a word in an encyclopaedia, especially as it is not always easy to specify exactly what it is that one wants to look up. Nevertheless, this “encyclopaedia” of mathematics is an incredible resource. And just as, if one were to try to get rid of all the entries in an encyclopaedia, or, to give a different comparison, all the books in a library, that nobody ever looked up, the result would be a greatly impoverished encyclopaedia or library, so, any attempt to purge mathematics of its less useful parts would almost certainly be very damaging to the more useful parts as well.
The Importance of Mathematics (Gowers)
For the past few days I’ve been pondering the question of how best to respond to paulfchristiano’s recent posts and comments questioning the value of mathematical research. I don’t think I can do it concisely, in a single post; bridging the inferential distance may require something more like a sequence of posts. I may end up writing such a sequence eventually, since it would involve ideas I’ve actually been wanting to write up for some time, and which are actually relevant to more than just the specific questions at issue here (whether society should sponsor mathematics, and given that it does, whether paulfchristiano or anyone else in the LW readership should pursue it).
However, as the preceding parenthetical hints at, I’m actually somewhat conflicted about whether I should even bother. Although I believe that mathematical research should be conducted by somebody, it’s not at all clear to me that the discipline needs more people beyond those who already “get” its importance, and are out there doing it rather than writing skeptical posts like paulfchristiano’s. It seems perfectly plausible to me that those who feel as paulfchristiano does should just leave the profession and do something else that feels more “important” to them. This is surely the best practical solution on an individual level for those who think they have a better idea than existing institutions of where the most promising research directions lie, at least until Hansonian prediction markets are (ever) implemented.
Nevertheless, for those interested in the society-level question of whether mathematics (as such) may be justifiably pursued by anyone, or any community of people, as a professional occupation (which is quite distinct from the question of whether e.g. paulfchristiano should personally pursue it), I recommend, at least as a start, grappling with the arguments put forward by the best mathematicians in their own words. I think this essay by Timothy Gowers (a Fields Medalist), titled “The Importance of Mathematics”, is a good place to begin. I would particularly draw the attention of those like paulfchristiano, who think they have a good idea of which branches of mathematics are useful and which aren’t, to the following passage, from pp.8-9 (unfortunately the illustrations are missing, but the point being made is pretty clear nonetheless):